Question 7 of 7: Multinomial Logit Mode Choice and the IIA Property

Question 7: Multinomial Logit Mode Choice and the IIA Property (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Three modes with mode-specific utility functions and the level-of-service data below; travel cost is in dollars.

Find. The three choice probabilities before and after the bus service improvement, and an assessment of the result against the independence-of-irrelevant-alternatives property.

Approach. Evaluate each observable utility, exponentiate, and normalise by the sum; then repeat with the improved bus attributes and compare the two share vectors against what IIA predicts.

1. Evaluate the observable utility of each mode (part a). Substituting the level-of-service data into the three utility functions, \[V_a = 0.1 - 0.02(12) - 0.15(7) - 0.03(2.5) = 0.1 - 0.24 - 1.05 - 0.075 = -1.2650,\] \[V_b = 0.2 - 0.03(20) - 0.15(12) - 0.03(0.75) = 0.2 - 0.60 - 1.80 - 0.0225 = -2.2225,\] \[V_r = -0.03(18) - 0.15(10) - 0.03(1.2) = -0.54 - 1.50 - 0.036 = -2.0760.\] Only differences in utility matter in a logit model, so the absolute values carry no interpretation; the automobile leads by about 0.81 utility units over light rail chiefly because its out-of-vehicle time is the smallest.
2. Exponentiate and normalise. The multinomial logit share is \(P_i = e^{V_i} / \sum_j e^{V_j}\), so \[e^{V_a} = 0.28224,\qquad e^{V_b} = 0.10834,\qquad e^{V_r} = 0.12543,\qquad \textstyle\sum_j e^{V_j} = 0.51600,\] \[P_a = \frac{0.28224}{0.51600} = 0.5470,\qquad P_b = \frac{0.10834}{0.51600} = 0.2099,\qquad P_r = \frac{0.12543}{0.51600} = 0.2431.\] \[\boxed{P_a = 54.7\%,\quad P_b = 21.0\%,\quad P_r = 24.3\%}\] The three sum to unity, which is the arithmetic check.
3. Re-evaluate the bus utility after the service improvement (part b). With in-vehicle time cut to 15 minutes and out-of-vehicle time to 10 minutes and the fare unchanged, \[V_b' = 0.2 - 0.03(15) - 0.15(10) - 0.03(0.75) = 0.2 - 0.45 - 1.50 - 0.0225 = -1.7725,\] an improvement of 0.45 utility units. Note where it comes from: the two-minute cut in waiting time is worth \(0.15 \times 2 = 0.30\) units against \(0.03 \times 5 = 0.15\) for the five-minute cut in riding time, because out-of-vehicle time is penalised five times as heavily as in-vehicle time.
4. Recompute the shares with the automobile and rail utilities unchanged. \[e^{V_b'} = 0.16991,\qquad \textstyle\sum_j e^{V_j} = 0.57758,\] \[P_a' = \frac{0.28224}{0.57758} = 0.4887,\qquad P_b' = \frac{0.16991}{0.57758} = 0.2942,\qquad P_r' = \frac{0.12543}{0.57758} = 0.2172.\] \[\boxed{P_a' = 48.9\%,\quad P_b' = 29.4\%,\quad P_r' = 21.7\%}\]
5. Quantify what moved. The bus gains 8.42 percentage points; the automobile loses 5.83 and light rail loses 2.59. The automobile therefore supplies 69.2 per cent of the bus’s gain and light rail 30.8 per cent — in exactly the ratio of their original shares, 54.70 to 24.31.

(c) Does the result make intuitive sense?

The direction is right; the split of the loss is not. That the bus gains share when its service improves is correct and reassuring. What is not credible is where the gain comes from, and the numbers make the problem visible rather than merely arguable. Because neither the automobile nor the light-rail utility changed, both were rescaled by the same factor when the denominator grew: \(0.51600 / 0.57758 = 0.8934\). Applying it, \(0.5470 \times 0.8934 = 0.4887\) and \(0.2431 \times 0.8934 = 0.2172\), reproducing both new shares exactly. Equivalently, the automobile-to-rail odds ratio is \[\frac{P_a}{P_r} = \frac{0.5470}{0.2431} = 2.2502 \qquad\text{and}\qquad \frac{P_a'}{P_r'} = \frac{0.4887}{0.2172} = 2.2502,\] identical to four decimal places. That invariance is the independence of irrelevant alternatives: in a multinomial logit model the ratio of the probabilities of any two alternatives depends only on those two alternatives’ own utilities, so a change to a third alternative must leave it untouched. By contrast, the automobile-to-bus odds ratio does change, from 2.6052 to 1.6612, because a bus attribute is what moved.

Why that is behaviourally wrong here. The bus and the light rail are close substitutes: both are public transit, both require walking to a stop and waiting, both serve the same corridor, and a traveller willing to use one is far more likely to switch to the other than a habitual car commuter is to abandon a car. A better bus should therefore draw disproportionately from light rail, not proportionally from both. This is the classic red-bus/blue-bus problem, and it arises because the logit error terms are assumed independent and identically distributed Gumbel — an assumption that ignores the unobserved attributes bus and rail share. The practical consequence for this corridor is that the model over-predicts the reduction in automobile trips and therefore over-states the congestion and emissions benefit of the bus investment, which is exactly the wrong error to make in a business case.

How to overcome it. The standard remedy is a nested logit model that places bus and light rail in a common transit nest under a single transit composite, with the automobile competing against that composite at the upper level; the nest inclusive-value parameter, which must lie between zero and one, measures how much substitution stays inside the nest, and the improved bus then draws mainly from rail before the transit total competes with the car. Where the nesting structure is not clean — a mode may plausibly belong to more than one group — a cross-nested or paired-combinatorial logit allows partial membership. Mixed (random-parameters) logit and multinomial probit relax the error assumption directly by allowing correlated and heterogeneous tastes, at the cost of simulation-based estimation. Cheaper partial fixes are also available: segmenting the market by car availability and income, since much of the apparent correlation is really unobserved heterogeneity; and adding explanatory variables that capture the shared attributes explicitly, such as a transfer penalty, a service-reliability term or a transit-specific constant. Whichever route is taken, the assumption should be tested rather than assumed away — the Hausman–McFadden specification test, which re-estimates the model on a restricted choice set and compares the coefficients, is the standard check, and it would almost certainly reject IIA for this three-mode set.